A 4-Approximation Algorithm for Guarding 1.5D Terrainsjking/papers/king_terrain_latin06.pdf ·...

Introduction to Terrain GuardingPrevious Work

PreliminariesOur 4-Approximation Algorithm

Future Work

A 4-Approximation Algorithmfor

Guarding 1.5D Terrains

James KingMcGill University

March 21, 2006

James King A 4-Approximation Algorithm for Guarding 1.5D Terrains



Future Work

Outline

Introduction to Terrain Guarding

Previous Work

Preliminaries

Our 4-Approximation Algorithm

Future Work




Future Work

What is a 1.5D Terrain?

I Also known as an x-monotone chain.

I The terrain intersects any vertical line at most once.

I No caves or overhangs.

I Points on the terrain ‘see’ each other if the line segmentconnecting them is never below the terrain.




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What is a 1.5D Terrain?

I Also known as an x-monotone chain.

I The terrain intersects any vertical line at most once.

I No caves or overhangs.

I Points on the terrain ‘see’ each other if the line segmentconnecting them is never below the terrain.




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The Terrain Guarding Problem

I We want a minimum set of guards that see the entire terrain.

I This is very similar to the Art Gallery Problem.




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The continuous problem:

I The entire terrain mustbe guarded.

I Guards can be placedanywhere.

I Closer to real-lifeapplications.

The discrete problem:

I Only vertices need to beguarded.

I Guards can only be placed onvertices.

I Closely related to non-geometricproblems (e.g. Vertex Cover).




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Outline


Previous Work

Preliminaries


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Is 1.5D Terrain Guarding NP-Complete?

I We don’t know.I Many related problems

are NP-complete:I Art Gallery Problem.I Vertex Domination.I Set Cover.I 2.5D Terrain Guarding.

I However, 1.5D Terrainsseem to forbid complexconstructions.




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Is 1.5D Terrain Guarding NP-Complete?

I We don’t know.I Many related problems

are NP-complete:I Art Gallery Problem.I Vertex Domination.I Set Cover.I 2.5D Terrain Guarding.

I However, 1.5D Terrainsseem to forbid complexconstructions.

Not 1.5D




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Terrain Guarding Algorithms

I Constant factor approximation algorithms given by:I Ben-Moshe, Katz, and Mitchell.I Clarkson and Varadarajan (randomized).

I They did not attempt to minimize the approximation factor.




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I Our contribution is a 4-approximation algorithm.

I Simpler than previous algorithms.

I Best approximation factor so far.

I We will present the algorithm for the discrete problem.

I Works for the continuous problem with a slight modification.

I Runs in Θ(n2) time.




Future Work

Outline


Previous Work

Preliminaries


Future Work




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Terminology and Notation

I p < q means p is to the left of q.

I p dominates q means p sees every unguarded point that qsees.

I L(p) is the leftmost point that sees p.

I R(p) is the rightmost point that sees p.




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Order Claim

I The fundamental property of 1.5D terrains that we exploit.

a

b

cd

a

b

cd

I Consider a < b < c < d .

I If a sees c and b sees d then a sees d .




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External Domination

I Consider a < b < c such that a sees c .

I {a, c} dominates b outside the interval (a, c).

a

b

c




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Outline


Previous Work

Preliminaries


Future Work




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The Algorithm

while unguarded points remain dofind u, S(u) such that:• u is unguarded• |S(u)| ≤ 4• S(u) dominates any guard that sees u

place guards at the points in S(u)end while

I This guarantees an approximation factor of 4.

I The real work is finding u and S(u).




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Finding u and S(u)

I The work is done by GuardRight, a recursive method.

I GuardLeft is the mirror image of GuardRight.

GuardRight

GuardLeft

GuardRight

recurse

recurse

place guards

I GuardRight recurses by callingGuardLeft and vice versa.

I Guards are placed only in theterminal case.




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GuardRight(p, c)

I p is an unguarded vertex (noton the convex hull).

I c sees every unguarded vertex in[L(R(p)), p)

I We look for u and S(u) in[L(R(p)), R(p)].

I We either find them or isolate asubregion to recurse on.




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Guards on the Left

I Left vertices (•) are on theconvex hull between p andL(R(p)).

I Right vertices (•) are between pand R(p).

I If we place a guard at c wedon’t need to consider placingguards in [L(R(p)), p) except onleft vertices.




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Yet More Terminology

I Unguarded vertices in [p,R(p))are either exposed or sheltered.

I Exposed vertices ( ) can see aleft vertex.

I Sheltered vertices ( ) cannotsee any left vertex.

I p is exposed.

I d is a special exposed vertexthat our algorithm finds.




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The Terminal Case

I L(d) sees every exposed vertex to the right of L′(d).I L′(d) is the leftmost right vertex that sees d .

I If {c , L(d), L′(d), R(d)} sees every unguarded vertex in[L′(d), R(d)] then the set dominates any vertex that sees d .

I This is the terminal case so we place guards.

I u ← d .

I S(u) ← {c , L(d), L′(d), R(d)}.




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The Recursive Case

I If not terminal, there must be asheltered vertex in [L′(d),R(p))

I Define q as the rightmost shelteredvertex.

I L(d) sees every unguarded vertex in(q, R(L(q))].

I We call GuardLeft(q, L(d)).

R(L(q))

d

L(d)

p

R(p)

q

GuardRight(p,c)

GuardLeft(q,L(d))

L(R(p))




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The Recursive Case

I We recurse on [L(q), R(L(q))], which is a proper subterrain of[L(R(p)), R(p)].

I Problem size shrinks, so we reach a terminal case eventually.

I We just repeat this whole process until the entire terrain isguarded.




Future Work

Outline


Previous Work

Preliminaries


Future Work




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Future Work

I Primary question: is 1.5D terrain guarding in P or is itNP-complete?

I Characterize visibility graphs of terrains.I The order claim isn’t the only tool we can use!I What other tools are available?




Future Work

Thank you!


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A 4-Approximation Algorithm for Guarding 1.5D Terrainsjking/papers/king_terrain_latin06.pdf ·...

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